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Monday, July 17, 2006

Rx: Stephen Wolfram’s New Kind of Science

For Mother’s Day in May 2002, my husband Harvey ordered Stephen Wolfram's book A New Kind Of Science for me. Harvey died a few days later and the book arrived after his memorial service. That summer I read the 1200-plus pages of this self-published tome, and I felt grateful to Mr. Wolfram as the book proved to be engaging, exciting, and helped to carry me through the aftermath of losing Harvey.

The British-born Mr. Wolfram has been educated at Eton, Oxford and Caltech, published his first scientific paper at 15, received his PhD in Theoretical Physics at 20 (a student of the Nobel Laureate Richard Feynman), won a MacArthur genius grant at 21 and then became independently wealthy through his software “Mathematica”. A New Kind of Science may have been the first science book since Darwin's The Origin of Species to sell out its first printing on its first day. It took him ten years to complete the book, during which Mr. Wolfram lived as a recluse, working all night long and running his software company for several hours during the day. Mr. Wolfram estimates that while writing the book, he typed 100 million keystrokes and moved his computer mouse more than 100 miles.

The reason this book’s message represents a paradigm shift is because of its potential for filling the gap in science related to the limitations of conventional mathematical equations which are unable to predict issues such as the shape that smoke from a cigarette is likely to take or the manner in which a forest fire will burn. At the heart of A New Kind of Science is Wolfram’s revolutionary idea to replace mathematical equations with algorithms and his assertion that the Universe is better explained on the basis of simple computer programs.

This is from The Economist: “Mr. Wolfram unashamedly compares the potential impact of his work to that of Sir Isaac Newton's “Principia Mathematica”, and suggests that his discoveries can answer long-standing puzzles in mathematics, physics, biology and philosophy, from the fundamental laws of nature to the question of free will.”

Wolfram’s work has its origin in cellular automata invented by the Hungarian physicist John von Neumann. Briefly, a complex system can be represented by using squares on graph paper that can be colored either black or white. Starting with one initial black square, a rule can be written which determines how its neighbors will be colored in each succeeding step forward. The properties of cellular automata are best explored using a computer which can generate thousands of iterations instantaneously. Wolfram found that with repeated iterations, complexity can arise out of simple rules.

At the center of Wolfram’s research was a quest for a new level of simplicity. To do this, he moved from a two-dimensional grid to the one-dimensional world of the line. Why one dimension? Because, like the Universe itself in the beginning, it is cellular automata in their most elemental form. If Wolfram could find complexity in one- dimensional cellular automata, the simplest construction imaginable, then he could find it anywhere. For years Wolfram worked through the night to determine the unfolding of hundreds of thousands of possible rules, typically going to bed around 5:00 a.m. and getting up in time for lunch. Most of the rules quickly devolved into predictable, endless patterns. He began to fear that he had been lured into one of science’s many dead ends. But then one night in May 1984, an epiphany: Wolfram realized his mistake. He had entered into the project with a predetermined idea of how nature worked, assuming that natural systems begin with randomness and move toward order. But, now, he asked himself, what if you turned the whole idea upside down? What if you began with ordered conditions and looked at which rules spun out greater complexity? Through a long night, Wolfram tore through all his past work, papers flying, looking for examples that would prove his new model. Finally, close to dawn, he found it: Rule 30, a pattern that grew more intricate and unpredictable with each step. It was stuffed with what mathematicians call "emergent effects": events that cannot be predicted in advance. From the simplest of parts, Wolfram had created infinite complexity. The aha! moment had arrived. "The Rule 30 automaton is the most surprising thing I've ever seen in science," Wolfram told London's Daily Telegraph. "'Even though it starts off from just one black cell, applying the same simple rule over and over again makes Rule 30 produce [an] amazingly complex pattern.

And again, from The Economist:

This was Mr. Wolfram's Eurekamoment: it suggested to him that complex systems in nature—be they weather systems, turbid fluid flow, a zebra's stripes or the human mind—might all be governed by small and simple sets of rules”. Wolfram’s critical realization was that “many very simple programs produce great complexity” leading to his “Principle of Computational Equivalence: that whenever one sees behavior that is not obviously simple—in essentially any system—it can be thought of as corresponding to a computation of equivalent sophistication.

With repeated iterations, cellular automata can produce pictures of great complexity (such as the one shown here).

Many critical problems are not solvable by conventional mathematics, for example, Newton’s Law of Universal Gravitation which only applies to two bodies in space, but dissolves into chaos with three bodies when the method of first integrals is applied. Wolfram’s “new” idea is that the Universe functions on the basis of an algorithm and not on the basis of mathematical equations. In fact, he confidently predicts, "Within 50 years, more pieces of technology will be created on the basis of my science than on the basis of traditional science. People will learn about cellular automata before they learn about algebra."

Implications for Biology: Theseus was able to slaughter the Minotaur because Ariadne gave him a golden thread that allowed him to retrace his steps out of the labyrinth. Wolfram has provided us with the means to solve many complex problems by retracing steps from complexity to simple algorithms. Since I am a cancer researcher, the sections in the book which deal with the implications of this new science for the highly complex biologic systems are the most fascinating for me, particularly Wolfram’s challenge to natural selection as the defining force in evolution. Darwin was able to collect an enormous amount of physical evidence to convince his contemporaries that all living organisms can be traced back to a single origin of life; that species evolve gradually through an accumulation of small changes; and that evolution occurs because there is genetic variation in every generation, and relatively few individuals survive and pass on their favorable genetic characteristics to the next generation. Even if certain genes confer only a very slight advantage, they gradually become more common over a long period of time. Thus, natural selection of favorable traits leads to both survival of the fittest and causes the species to adapt to natural environments. The neo-Darwinists like Dawkins, Maynard Smith and Dennett believe that natural selection is the only driving force in evolution, which occurs gradually over a long period of time. Gould and Eldridge who consider themselves as pluralists (as opposed to the neo-Darwinists who are reductionists) on the other hand, suggest that evolving species should be viewed as complex systems and that evolution is not a slow, gradual process but occurs in short bursts when new species appear, followed by a long period of stasis (punctuated equilibrium). Finally, they also maintain that organisms can often have traits which have no apparent useful purpose, but appear as a mere by-product of evolutionary changes (“spandrels”). In fact, many very important characteristics of humans such as reading and writing are examples of such “emergent properties” since the brain had become large prior to existence of written language. In this debate, Wolfram’s new science supports neither Darwin nor Gould:

One of the most esteemed documents of modern paleontology is Stephen Jay Gould's doctoral thesis on shells. According to Gould, the fact that there are thousands of potential shell shapes in the world, but only a half dozen actual shell forms, is evidence of natural selection. Not so, says Wolfram. He's discovered a mathematical error in Gould's argument, and that, in fact, there are only six possible shell shapes, and all of them exist in the world. In other words, you don't need natural selection to pare down evolution to a few robust forms. Rather, organisms evolve outward to fill all the possible forms available to them by the rules of cellular automata. Complexity is destiny—and Darwin becomes a footnote. "I've come to believe," says Wolfram, "that natural selection is not all that important." The more sciences he probes, the more Wolfram senses a deeper pattern—an underlying force that defines not only the cosmos but living things as well: "Biologists," he says, "have never been able to really explain how things get made, how they develop, and where complicated forms come from. This is my answer." He points at the shell, "This mollusk is essentially running a biological software program. That program appears to be very complex. But once you understand it, it's actually very simple. --Forbes

In order to illustrate how complexity in biology arose from simple programs, I will summarize a few of the salient findings from the book here. (ANKS pp383-428)

In the past, the idea of optimization for some sophisticated purpose seemed the only conceivable explanation for the level of complexity in biology. Take the example of fish which manifest so many beautiful colors. One Darwinian explanation would be that these colors improved survival through either allowing the creatures to evade being hunted by blending with the environment or shocking the predator with their brilliant, contrasting colors. In other words, every one of these colors and patterns evolved for a purpose. But then why do all the multi-colored fish have the same exact internal structures while having such radically different patterns on the outside? Wolfram’s beguilingly simple explanation is that the most visually striking color differences have almost nothing to do with natural selection, but are reflections of completely random changes in underlying genetic programs. On the other hand, the vital features such as the internal organs have changed only quite slowly and gradually in the course of evolution because those are precisely the ones molded by natural selection.

“The range of pigmentation patterns on mollusc shells (picture) correspond remarkably closely with the range of patterns that are produced by simple randomly chosen programs based on cellular automata. There are already indications that such programs are quite short. One of the consequences of a program being short is that there is little room for inessential elements and any mutation or change in the program, however small, will tend to have a significant effect on at least the details of patterns it produces. Or biologic systems should be capable of generating essentially arbitrary complexity by using short programs formed by just a few mutations” (ANKS).

“But if complexity is this easy to get, why is it not even more widespread in biology? The answer is natural selection, which can achieve little when confronted with complex behavior. There are several reasons for this. First, with more complex behavior, there are huge numbers of possible variations. Second, complex behavior inevitably involves many elaborate details, and since different ones of these details may happen to be the deciding factors in the fates of individual organisms, it becomes very difficult for natural selection to act in a consistent and definitive way. Third, whenever the overall behavior of a system is more complex than its underlying program, almost any mutation in the program will lead to a whole collection of detailed changes in the behavior, so that natural selection has no opportunity to pick out changes which are beneficial from those which are not” (ANKS).

In the picture here, one can see the shell shapes generated by the simple model and those actually found in nature. The computerized array shows systematic variations of just two parameters.

Progress in science occurs when a new order is discovered which can unite seemingly unrelated facts. ANKS has achieved just that by showing that it is possible to get complexity from simple programs in any system. The job of a scientist in the conventional sense is to understand how things work and not why. Wolfram has combined the two by asking not only how evolution occurs, but also the meaning of it all. And the answer is terribly disturbing. No meaning at all. “So why do higher organisms exist at all? My guess is that it has almost nothing to do with optimality, and that instead it is essentially just a consequence of strings of random mutations that happened to add more and more features without introducing fatal flaws” (ANKS p398).

Richard Dawkins observed, “Darwin made it possible to be an intellectually fulfilled atheist”. But for the theists, Wolfram has changed the concept of God from being a mathematician to being a software engineer.

Dr. Stephen Wolfram will be the honored guest speaker at the 4th Harvey Preisler Memorial Lecture to be held at the University of Massachusetts in Worcester on July 28, 2006.